The set of real values of $α$ for which the system of linear equations
$x + (\sin α) y + (\cos α) z = 0,$
$x + (\cos α) y + (\sin α) z = 0,$
$- x + (\sin α) y - (\cos α) z = 0$
has a non trivial solutions is

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$\frac{n π}{2} + {(- 1)}^{n} \frac{π}{8} : n \in z$

$\frac{n π}{2} + {(- 1)}^{n} \frac{π}{4} + \frac{π}{8} : n \in z$

$\frac{n π}{2} + {(- 1)}^{n} \frac{π}{4} - \frac{π}{8} : n \in z$

$\frac{n π}{2} + {(- 1)}^{n} \frac{π}{8} - \frac{π}{8} : n \in z$

answer is D.

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Detailed Solution

$D e t = 0 \Rightarrow \sin 2 α + \cos 2 α = 1$

$\Rightarrow \sin (2 α + \frac{π}{4}) = \frac{1}{\sqrt{2}} = \sin \frac{π}{4}$

$\Rightarrow 2 α = n π + {(- 1)}^{n} \frac{π}{4} - \frac{π}{4} : n \in z$

$\Rightarrow α = \frac{n π}{2} + {(- 1)}^{n} \frac{π}{8} - \frac{π}{8} : n \in z$

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